The present invention relates to a method and a apparatus for the free-form optimization of bevel and hypoid gears, to methods and apparatus for the production of thus optimized gears and to a method and a apparatus for the correction of already produced bevel and hypoid gears by means of free-form optimization.
In the manufacture of gears, in particular the manufacture of bevel and hypoid gears, the running properties and the stressing of such gears, such as, for example, their quiet running, nowadays have to satisfy ever higher requirements. These properties can no longer be achieved by means of gears, such as can be manufactured, for example, on machines of an older type of construction, since these machines do not afford sufficient possibilities for influencing the surface geometry of the gears in the light of modern increased requirements regarding running properties and stress.
To be precise, in these machines of an older type of construction, referred to below as non-free-form machines, in order to influence the gearwheel geometry, in particular the tooth geometry, there is in the first place only a selection of what are known as basic machine settings available which each have a theoretical significance for the gearwheel to be produced. These basic machine settings in this case relate to what is known as a basic machine which constitutes a mathematically theoretical model in relation to which all conceivable real non-free-form machines can be uniquely mapped in a reversible way, the real non-free-form machines mostly having fewer axes than the basic machine (also referred to as a basic tooth-cutting machine), but even then, not all types of toothings can be manufactured.
An illustration of this basic machine and its basic machine settings is found, for example, in Goldrich (Goldrich, “CNC Generation of Spiral Bevel and Hypoid Gears: Theory and Practice”, The Gleason Works, Rochester, N.Y. 1990), although the significance of these is also discussed in EP 0 784 525. Admittedly, the non-free-form machines of an older type of construction, which are based on such a basic machine, have been modified, in the course of technical development, such that, in addition to the fixed axis settings mentioned, they also allowed certain (additional) movements, such as modified roll and/or helical motion, but EP 0 784 525 also states why, at least in the view put forward there, the possibilities for surface geometry optimization during the use of a basic machine are nevertheless inadequate.
However, in the meantime, gearwheel production machines have also been developed, which reduce the number of machine axes, as compared with a basic machine, to the necessary extent, so that a tool can be oriented in terms of a workpiece in such a way that a gearwheel can be produced. These machines have linear and pivot axes and axes of rotaion instead of the basic machine axes necessary for the basic machine settings, in order to ensure the necessary degrees of positioning freedom for gearwheel production, and are also referred to as free-form machines (cf., in this respect, likewise the statements in EP 0 784 525, and also U.S. Pat. No. 4,981,402). Conventionally, such free-form machines have up to six machine axes, specifically, preferably, three linear axes and three axes of rotation, thus constituting a marked simplification in mechanical terms, as compared with a non-free-form machine based on the model of a basic machine.
However, free-form machines of this type not only have a significantly simpler construction than non-free-form machines in mechanical terms, but, furthermore, also afford a further fundamental advantage: with regard to the degrees of freedom made possible in this case, the machining of the workpiece is no longer tied to the, now historic, rigid theoretical significance of the basic-machine axes for the gearwheel, but, instead, is, in principle, completely free and thus also makes it possible, as compared with the basic machines, to generate more sophisticated gearwheel surface geometries, by means of which, for example, the requirements as regards the gearwheel running properties and gearwheel stress, which were mentioned in the introduction and have risen markedly in comparison with earlier times, can fundamentally be achieved.
For the free-form machines, too, there is in this case a theoretical model, to be precise a free-form basic machine with at most six axes, which has a gearwheel to be machined and a tool which can in each case be rotated about an axis, and the tool and the gearwheel to be machined are moveable, preferably displaceable or rotatable, with respect to one another along or about a plurality of axes.
As regards terminology, the foregoing is to be taken as a summary and, for what follows, it is also to be stated, in advance, that, here, this publication in each case designates                basic machine: as a model of non-free-form machines, in relation to which all conceivable real non-free-form machines can be uniquely mapped in a reversible way, even to symmetries, and in which each machine axis has a significance in toothing theory,        non-free-form machine: as a real machine of an older type of construction, in which a selection of what are known as basic machine settings of the basic machine is available, which in each case have a theoretical significance for the gearwheel to be produced, but in which all the machine axes of the basic machine do not necessarily even actually have to be present in real terms,        free-form basic machine: as a model of free-form machines with at most six axes, which has a gearwheel to be machined and a tool which are in each case rotatable about an axis, and the tool and the gearwheel to be machined are moveable, preferably displaceable or rotatable, with respect to one another along or about a plurality of axes and can be uniquely mapped in relation to the free-form machines in a reversible way, even to symmetries, and        free-form machine: as a real machine which has linear and pivot axes and axes of rotation instead of the axes of the basic machine which are necessary for the basic machine settings, in order to ensure the necessary degrees of positioning freedom for gearwheel production (these having conventionally, but not necessarily, up to six machine axes, specifically, preferably three linear axes and three axes of rotation).        
In particular, in this connection, it is to be pointed out that the term “basic machine” is used only in the above sense and not, for example, as a generic term (in particular, not even as a generic term for free-form basic machines). Furthermore, here and further on, mapping between machines is understood as meaning the mapping of all possible movements of one machine in relation to movements of the others, that is to say not just the mapping of an actual implementation of a production method.
Thus, for example, the prior art according to Krenzer (Krenzer, T., “Neue Verzahnungsgeometrie für Kegelräder durch Schleifen mit kegeligen Topfscheiben [‘Flared Cup’ Verfahren]” [“New Toothing Geometry for Bevel Wheels by Grinding with Conical Cup Wheels [‘Flared-Cup’ method]”] in: Theodore J. Krenzer: CNC Bevel Gear Generators and Flared Cup Formate Gear Grinding. The Gleason Works, Rochester, N.Y., 1991) fundamentally discloses the influence which a suitable control of the axes of such a free-form machine can exert, for example, on the contact-pattern improvement of a gearwheel (a property essentially determining the running properties of a gearwheel). However, this flared-cup method is a straightforward forming method in which use is made of the fact that the special tool touches the workpiece wheel in only one line in the profile height direction. This made it possible as it were to “test” the simple monocausal relations, given here, of the forming process in terms of the effect of the respective variation in the control of only a single axis in each case.
The prior art also makes use of this fundamental finding in that it attempts to use these additional degrees of freedom for correcting the surface geometry of bevel or hypoid gears, for example by means of the method for production of tooth-flank modifications according to EP 0 784 525.
A way of tooth-flank modification of the gears in question here is proposed in EP 0 784 525. In more general terms, this presents a method for modifying the surface geometry of such gears, in which, first, the basic machine settings to be originally selected only fixedly for the entire production process in terms of their value are replaced by basic machine setting functions, whereby the value of the basic machine setting can vary during the process of producing the gearwheel, as a result of which, according to the statements of this publication, it is to be possible to modify in a targeted manner the tooth flank and consequently the surface geometry of the gearwheel. This is to take place such that the functional variations described above are converted on the theoretical basic machine there, by means of a method known, for example, from U.S. Pat. No. 4,981,402, to a free-form machine which then carries out the actual gearwheel generation.
According to the statements in EP 0 784 525, therefore, this procedure affords the possibility of conducting gearwheel design calculations and considerations of theoretical significance for the gearwheel, which are based on the theoretical basic machine model, but at the same time, likewise according to the presentation there, also of providing additional freedoms in gearwheel development, in that all the basic machine axes, which could previously be set only fixedly (statically), are used as functionally variable (active) settings during the process of producing the gearwheel, the intention of this being to allow the abovementioned model of the basic machine and its transformation to a free-form machine.
The additional possibilities, fundamentally afforded according to the already mentioned article by Krenzer (see above), and in any case relating only to the special “Flared-Cup method”, of the free-form machine for optimizing the surface geometry of a bevel or hypoid gearwheel are thus, according to EP 0 784 525, to be utilized by way of a theoretical basic machine with basic machine settings functionally variable during the production process and by way of its subsequent transformation to the free-form machine.
This procedure has the disadvantage, however, that it uses a multiplicity of redundant control parameters for optimization in the form of the coefficients of the functions mapping the axial movements, since the theoretical basic machine always has more axes than the free-form machine. The result of this is that, because of these parameters, optimization is unnecessarily difficult as a result of these redundancies. Thus, for example when numerical optimization methods employing Jacobi matrices are used, in the case of such redundant parameters there are regularly linear dependences, leading to singular Jacobi matrices, thus resulting in an optimization task which is substantially more difficult to master in numerical terms (cf., also, Nocedal, J. and Wright, S. J., “Numerical Optimization”, Springer Series in Operations Research, New York, 1999) than if it were based on regular Jacobi matrices, such as occur when non-redundant parameters are used.
This procedure therefore has room for improvement in view of the significance of the optimization method for gears which require the said optimization of their surface geometry, for example in terms of the currently markedly increased requirements, already mentioned in the introduction, with regard to the running properties; this is also particularly because the original theoretical significances of the machine axes no longer have such importance with regard to this optimization of the surface geometry and may therefore be dispensed with in favour of an improvement in optimization.
For such an improvement, however, a method would be required which makes it possible to carry out optimizations of the surface geometry of bevel or hypoid gears directly on the free-form machine, that is to say, in particular, without going by way of the model of the basic machine, in order thereby to avoid the abovementioned disadvantages. Such a method has not hitherto been known according to the prior art.
On the contrary, the methods employed hitherto according to the prior art are all based on the basic machine with its basic machine settings which in each case have a theoretical significance for the gearwheel to be produced:
Thus, for example, findings in toothing theory, with investigations by simulation of, for example, the influence of the parameters on the flank form or the ease-off (cf., also, Wiener, D., “Örtliche 3D-Flankenkorrekturen zur Optimierung spiralverzahnter Kegelräder” [“Local 3D Flank Corrections for the Optimization of Helically Toothed Bevel Wheels”] in: Seminar Documentation “Innovationen rund urns Kegelrad” [“Innovations around the Bevel Wheel”], WZL, RWTH Aachen 2001, but also Stadtfelt, H. J., “The Universal Motion Concept for Bevel Gear Production”, in: Proceedings of the 4th World Congress on Gearing and Power Transmission, Volume 1, Paris 1999, pp. 595-697) or, for example, the influence on load contact properties (cf., also, Simon, V., “Optimal Machine Tool Setting for Hypoid Gears Improving Load Distribution”, ASME Journal of Mechanical Design 123, December 2001, pp. 577-582) are combined, in order to derive suitable algorithms for the configuration of the gearwheel surface geometry.
Also, according to the prior art (cf., for example, Gosselin, C.; Guertin, T.; Remond, D. and Jean, Y., “Simulation and Experimental Measurement of the Transmission Error of Real Hypoid Gears Under Load”, ASME Journal of Mechanical Design Vol. 122, March 2000, or Gosselin, C.; Masseth, J. and Noga, S., “Stock Distribution Optimization in Fixed Setting Hypoid Pinions”, Gear Technology July/August 2001), a Newton-Raphson method is employed in order to carry out flank corrections or special flank adaptations, hence gearwheel geometry modifications or optimizations. However, in this sensitivity-based optimization, only the fixed basic machine settings are varied, in order to achieve flank-form modifications at most of 2nd order.
Probably the latest development in the field of the application of modern optimization techniques for gearwheel optimization is the publication “Automatisches Differenzieren im Maschinenbau-Simulation und Optimierung bogenverzahnter Kegelradgetriebe” [“Automatic Differentiation in Mechanical Engineering-Simulation and Optimization of Spirally Toothed Bevel Wheel Transmissions”] by 0. Vogel et al. (Vogel, O., Griewank, A., Henlich, T. and Schlecht, T., “Automatisches Differenzieren im Maschinenbau-Simulation und Optimierung bogenverzahnter Kegelradgetriebe” [Automatic Differentiation in Mechanical Engineering-Simulation and Optimization of Spirally Toothed Bevel Wheel Transmissions”], in: Conference Volume “Dresdner Maschinenelemente Kolloquium-DMK2003” [“Dresdner Machine Element Conference-DMK2003”], TU Dresden 2003, pp. 177-194).
As already mentioned, however, all the aforesaid methods always relate to the control of the basic machine with its basic machine settings and their respective significances in toothing theory. This is to be explained against the historic technical background of the procedure of the engineers designing gears who in this case aim at exactly these variables with gearwheel-theoretical significance. Such a procedure is also perfectly useful for the basic design of gears, that is to say those gears which have no modifications in their geometry, as compared with the basic forms thereby achievable, since in this case, on the one hand, the theoretical relation originating from the imagination of a person skilled in the art continues to exist, and, on the other hand, the above-mentioned disadvantages do not occur.
For the use of the numerical optimization methods, however, contrary to the opinion according to the prior art (cf., inter alia, EP 0 784 525 B1, there, for example, paragraph number [0016] and [0023]), this toothing-theoretical significance is ultimately completely irrelevant; this is if only because, simply due to the multiplicity of parameters, a person skilled in the art can no longer conclude, from the result of any optimization taking place on a free-form basic machine, what is actually taking place in the production process as a result of these parameters. A toothing-theoretical reference of the machine model used is therefore inconsequential if only for this reason.